The Non-linear Schr\(\ddot{o}\) dinger equation(NLSE) appears in the context of mathematical modelling of many physical
phenomena in different domains of science, like optics in nonlinear media [1]}, [2]}, [3]}, Bose-Einstein
Condensates(BEC)[4]}, [5]}, [6]}, plasma physics[7]}, gravity waves[8]},
Bio-molecular dynamics[9]} etc. NLSE exhibits
soliton[10]}, [11]}, [12]}, [13]} solutions and is an integrable system with a very rich
mathematical structure[14]}, [15]}, [16]}, [17]}. The quantized NLSE describes a hardcore Bose-gas with
delta-function interaction for which various physical properties including correlations functions
can be computed analytically[18]}.
Several generalisations of the NLSE have been considered to model different emerging phenomena
in physics over the last few decades. For example, the NLSE with an external confining potential
is known as Gross-Pitaevski equation[5]} and is relevant in the context of BEC. The investigations
on collapse and growth of quasi-condensates in NLSE with harmonic confinement offers a rich variety
of mathematical structure[20]}, [21]}, [22]}. Localized, unbounded and periodic potentials have also been investigated
with interesting results. The NLSE with space[23]}, [24]}, [25]}, [26]}, [27]}, [28]}, [29]} and
time[30]}, [31]}, [32]}, [33]}, [34]}, [35]}, [36]}, [37]}, [38]}, [39]}, [40]}, [41]}
modulated nonlinear strengths is also of immense interest in the same context.
